Zassenhaus lemma
E827064
The Zassenhaus lemma is a fundamental result in group theory that describes how subgroups in a group extension correspond and relate to each other, often used in the study of composition series and the Jordan–Hölder theorem.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Zassenhaus lemma canonical | 2 |
| butterfly lemma | 2 |
| Zassenhaus lemma in group theory | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T9867904 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Zassenhaus lemma Context triple: [Hans Zassenhaus, notableWork, Zassenhaus lemma]
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A.
Schreier refinement theorem
The Schreier refinement theorem is a result in group theory stating that any two subnormal series of a group admit equivalent refinements, serving as a precursor and companion to the Jordan–Hölder theorem.
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B.
Jordan–Hölder theorem
The Jordan–Hölder theorem is a fundamental result in group theory stating that any two composition series of a finite group have the same length and the same (up to order and isomorphism) simple factor groups.
-
C.
Noether's isomorphism theorems
Noether's isomorphism theorems are fundamental results in abstract algebra that relate quotient structures and substructures of groups, rings, and modules, providing a unifying framework for understanding homomorphic images and factor structures.
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D.
Cauchy's theorem in group theory
Cauchy's theorem in group theory is a fundamental result stating that if a finite group’s order is divisible by a prime p, then the group contains an element (and hence a subgroup) of order p.
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E.
Eilenberg–Zilber theorem
The Eilenberg–Zilber theorem is a fundamental result in algebraic topology that establishes a chain homotopy equivalence between the singular chain complex of a product space and the tensor product of the singular chain complexes of the factors.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Zassenhaus lemma Target entity description: The Zassenhaus lemma is a fundamental result in group theory that describes how subgroups in a group extension correspond and relate to each other, often used in the study of composition series and the Jordan–Hölder theorem.
-
A.
Schreier refinement theorem
The Schreier refinement theorem is a result in group theory stating that any two subnormal series of a group admit equivalent refinements, serving as a precursor and companion to the Jordan–Hölder theorem.
-
B.
Jordan–Hölder theorem
The Jordan–Hölder theorem is a fundamental result in group theory stating that any two composition series of a finite group have the same length and the same (up to order and isomorphism) simple factor groups.
-
C.
Noether's isomorphism theorems
Noether's isomorphism theorems are fundamental results in abstract algebra that relate quotient structures and substructures of groups, rings, and modules, providing a unifying framework for understanding homomorphic images and factor structures.
-
D.
Cauchy's theorem in group theory
Cauchy's theorem in group theory is a fundamental result stating that if a finite group’s order is divisible by a prime p, then the group contains an element (and hence a subgroup) of order p.
-
E.
Eilenberg–Zilber theorem
The Eilenberg–Zilber theorem is a fundamental result in algebraic topology that establishes a chain homotopy equivalence between the singular chain complex of a product space and the tensor product of the singular chain complexes of the factors.
- F. None of above. chosen
Statements (33)
| Predicate | Object |
|---|---|
| instanceOf |
group theory lemma
ⓘ
mathematical theorem ⓘ |
| alsoKnownAs |
butterfly lemma
ⓘ
linked to:
Zassenhaus lemma
|
| appearsIn |
algebra textbooks
ⓘ
group theory monographs ⓘ |
| appliesTo |
groups
ⓘ
subgroups ⓘ |
| category |
theorems about finite groups
ⓘ
theorems about series of subgroups ⓘ |
| concerns |
correspondence between certain factor groups
ⓘ
isomorphisms between quotient groups ⓘ |
| describes | relations between subgroups in a group extension ⓘ |
| field | group theory ⓘ |
| hasDiagrammaticNickname | butterfly diagram ⓘ |
| hasGeneralization | results on modular lattices of subgroups ⓘ |
| hasRole | technical tool in the proof of Jordan–Hölder theorem ⓘ |
| implies | existence of isomorphisms between certain factor groups in two subgroup series ⓘ |
| namedAfter | Hans Zassenhaus ⓘ |
| relatedTo |
Jordan–Hölder theorem
ⓘ
Schreier refinement theorem ⓘ isomorphism theorems for groups ⓘ
linked to:
Noether's isomorphism theorems
|
| relatesConcept |
group extensions
ⓘ
normal subgroups ⓘ refinements of series of subgroups ⓘ subgroup lattices ⓘ |
| statedInTermsOf |
intersections of subgroups
ⓘ
products of subgroups ⓘ |
| typicalContext |
development of the Jordan–Hölder theorem
ⓘ
discussion of composition series and chief series ⓘ |
| usedFor |
comparing different subnormal series
ⓘ
refining series of subgroups ⓘ |
| usedIn |
proofs of the Jordan–Hölder theorem
ⓘ
study of composition series ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Zassenhaus lemma Description of subject: The Zassenhaus lemma is a fundamental result in group theory that describes how subgroups in a group extension correspond and relate to each other, often used in the study of composition series and the Jordan–Hölder theorem.
Referenced by (5)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Zassenhaus lemma
linked to: Zassenhaus lemma
linked to: Zassenhaus lemma